Question

A recipe for cookies calls for \(\frac{3}{8}\) cup of brown sugar to make 2 dozen cookies. If Sasha has \(4.5\) cups of brown sugar available and enough of all the other ingredients, how many dozen cookies can he make? (A) 9 dozen (B) 12 dozen (C) 15 dozen (D) 24 dozen

Short answer

Sasha can make 24 dozen cookies with 4.5 cups of brown sugar. The correct answer is (D) 24 dozen.

Step-by-step solution

Set up the proportion

We can set up the proportion by putting the amount of brown sugar in the numerator and the number of dozens of cookies in the denominator \[ \frac{3/8 \text{ cup}}{2 \text{ dozen}} = \frac{4.5 \text{ cup}}{x \text{ dozen}} \]

Solve for x

To solve for x, we cross-multiply and then isolate x \[ \frac{3}{8}x = 2(4.5) \]

Calculate x

Now let's calculate the value of x \[ x = \frac{2(4.5)(8)}{3} \] Now we have: \[ x = \frac{(9)(8)}{3} \] Divide both numbers by 3 \[ x = \frac{(3)(8)}{1} \] Multiply 3 by 8 \[ x = 24 \] So, Sasha can make 24 dozens of cookies with 4.5 cups of brown sugar. The correct answer is (D) 24 dozen.

Key concepts

Proportions in Math

Proportions are one of the fundamental concepts in mathematics, especially when dealing with real-world problems. A proportion is an equation that states two ratios are equivalent. For instance, if a recipe calls for a certain amount of an ingredient relative to a portion of food, and you want to scale up this recipe, you would use proportions to ensure the flavors remain balanced.

Let's consider the example from the original exercise with cookies. The ratio here is \( \frac{3/8\text{ cup of brown sugar}}{2\text{ dozen cookies}} \), which helps us understand that for every 3/8 cup of sugar, 2 dozen cookies can be made. This balance is crucial to maintain when scaling the recipe up or down in order to get the same taste and texture that the original recipe intends.

When working with proportions, there are a few key things to remember:

• Proportions are about comparing parts to a whole.
• They help us solve problems involving scale or ratio.
• They are often represented using fractions or ratios, and can be solved by finding the cross-products.

Understanding proportions is an essential skill, particularly in cooking, construction, art, and any field where scaling is necessary.

Solving Proportions

Solving proportions is a matter of finding the missing term in a set of equivalent fractions or ratios. As seen in the original exercise, the process can be broken down into manageable steps. The first step is to set up a proportion equation that relates the quantities given in the problem. In our example, this was represented as \( \frac{3/8 \text{ cup}}{2 \text{ dozen}} = \frac{4.5 \text{ cup}}{x \text{ dozen}}\).

Cross-Multiplication

After setting up the proportion, we move on to solving for the unknown quantity, often denoted by 'x'. This typically involves cross-multiplication, where we multiply the numerator of one ratio by the denominator of the other ratio and set it equal to the product of the remaining numerator and denominator. This step eliminates the fractions and makes it easier to solve for 'x'.

Isolating the Variable

The next step is to isolate the variable. After cross-multiplying, you'll end up with a simple algebraic equation, which can be solved by performing basic arithmetic operations. In the final step of the original problem, dividing and then multiplying gives us the number of dozens of cookies Sasha can make, which is 24 dozen.

Solving proportions can feel daunting at first, but with practice, the steps become a straightforward way to find relationships between different quantities.

GED Math Practice

Preparing for the GED (General Educational Development) tests requires practice across various mathematical topics, including proportions. Proportions are practical and commonly featured in GED math problems because they reflect real-life situations like cooking, shopping, and construction, where mathematical reasoning is needed.

GED math practice should include solving many different kinds of proportions to build confidence and proficiency. Practice problems may vary from simple, direct proportions to more complex ones involving additional steps or concepts. For example, after solving a proportion you might need to use the result in a further calculation, just like converting the number of dozens into the number of individual cookies if that were asked.

Here are some tips for effective GED math practice:

• Start with simple examples to build foundational skills.
• Gradually progress to more complex problems.
• Time yourself to get used to the exam pressure.
• Review your solutions thoroughly to understand any mistakes.
• Practice consistently to improve speed and accuracy.

Working on these skills will not only prepare you for the GED but also for any situation where you need to apply mathematical reasoning in your daily life.